Geometry Seminar

                                                                                                                      Vanderbilt University
                                                                                                                             Fall  2015

   Organizers:  Gennadi Kasparov, Ioana Suvaina, Rares Rasdeaconu

   Fridays, 3:10-4:00pm in SC 1310 (unless otherwise noted)

      Friday, September 25th

Speaker: Tsuyoshi Kato, Kyoto University

Title: K-theoretic degree of the covering monopole map

Abstract: I will present a construction of K-theoretic degree of the covering monopole map as a homomorphism between
full group C^* algebras.

Friday, October 9th

Speaker: Rudy Rodsphon, Vanderbilt University

Title: Methods of cyclic cohomology in index theory

Abstract: The aim of this talk will be to review results in classical index theory through the point of view of cyclic
cohomology, developed by Alain Connes as an alternative of de Rham homology in Noncommutative Geometry.
We will then recall how it can be used to extend index theory beyond the classical setting. As an introduction to a second
talk in two weeks, it will be elementary and accessible (I hope) to graduate students. In particular, it should not contain
recent results.

Friday, October 23rd

Speaker: Rudy Rodsphon, Vanderbilt University

Title: On Connes-Moscovici transverse index problem

Abstract: This talk will be an independent continuation of a previous talk two weeks ago. We will sketch how
zeta functions and excision in cyclic cohomology may be combined to obtain equivariant index theorems
for a certain class of hypoelliptic operators arising naturally on foliations, actions being not necessarily proper.
As a corollary, we obtain a solution to a conjecture of Connes and Moscovici, on the calculation of index classes
of transversally elliptic operators on foliations (without holonomy). This is a joint work with Denis Perrot.

Friday, October 30th

Speaker: Anna Marie Bohmann, Vanderbilt University

Title: The Equivariant Generating Hypothesis

Abstract: Freyd's generating hypothesis is a long-standing conjecture in stable homotopy theory.  The conjecture
says that if a stable map between finite CW complexes induces the zero map on homotopy groups, then it must
actually be nullhomotopic.  I will formulate the appropriate generalization of this conjecture in the case where
a group G acts on the complexes and give some results about this setting.  In particular, I will show that the rational
version of this conjecture holds when G is finite, but fails when the group is S^1. 

Friday, November 6th

Speaker: Anna Marie Bohmann, Vanderbilt University

Title:  Constructing equivariant spectra

Abstract: Equivariant spectra determine cohomology theories that incorporate a group action on spaces.
Such spectra are increasingly important in algebraic topology but can be difficult to understand or construct.
I will discuss recent work with Angelica Osorno, in which we build such spectra out of purely algebraic data
based on symmetric monoidal categories. Our method is philosophically similar to classical work of Segal
on building nonequivariant spectra.

Friday, November 13th

Speaker: Spencer Dowdall, Vanderbilt University

Title:  Surface bundles, Teichmuller space, and mapping class groups

Abstract: This talk will introduce the mapping class group and Teichmuller space of a surface with a
focus on how these objects are related to the theory of surface bundles. We'll take the perspective of
Teichmuller space as a (sort of) classifying space for surface bundles and explain how each surface bundle
gives rise to a monodromy representation into the mapping class group. I'll then describe how the geometry
of Teichmuller space is related to the metric properties of surface bundles, which will lead us to the notion
of convex cocompact subgroups of mapping class groups. The talk will be introductory (and I hope
accessible!) in nature with the aim of setting the stage for a follow-up talk discussing some of my work
in this area.

Friday, November 20th

Speaker: Spencer Dowdall, Vanderbilt University

Title:  Hyperbolicity of surface group extensions, and  convex cocompact subgroups of mapping class groups

Abstract:  Convex cocompact subgroups of mapping class groups, as introduced by Farb and Mosher, are subgroups
whose action on Teichmuller space is analogous to that of convex cocompact Kleinian groups acting on hyperbolic
3-space. Moreover, it is exactly the convex cocompact subgroups that give rise to Gromov hyperbolic surface
bundles and to hyperbolic extensions of free groups. In this talk I will describe a setting, arising from hyperbolic
fibered 3-manifolds, in which there is a concrete connection between these two notions of convex cocompactness
and explain how one may use this connection to prove certain subgroups of mapping class groups are convex cocompact.
This is joint work with Richard Kent and Christopher Leininger.

Wednesday, December 2, 2015 (joint with the Topology & Group Theory Seminar)

Speaker: Rares Rasdeaconu, Vanderbilt University

Title: Rational curves on real K3 surfaces 

Abstract: The simply connected complex surfaces with vanishing first Chern class, also known as K3 surfaces, are
discussed from the perspective of their enumerative geometry. Do such surfaces contain rational curves, i.e.
holomorphically immersed 2-spheres? If yes, how many? Moreover, if the K3 surface admits an anti-holomorphic
involution, how many of these curves are invariant? I will try to give some answers to such questions based on recent
joint work with V. Kharlamov.

Friday, December 10th

Speaker: Ioana Suvaina, Vanderbilt University

Title:  Asymptotically locally Euclidean Kahler surfaces, analytic compactifications and classifications

                  Abstract:  In this talk, I will present the current state of the art of the study of ALE Kahler surfaces. I will give the classification
                  of ALE Ricci-flat Kahler surfaces, and I propose a classification of ALE scalar flat Kahler surfaces. The new techniques employed
                  are based on analytic compactifications and deformation theory.


 Old Seminar Web-Pages: Fall 2009, Fall 2010, Spring 2011, Fall 2011, Spring 2012, Fall 2012, Spring 2013, Fall 2013, Spring 2014, Fall 2014,